College Algebra Exam Review 350

College Algebra Exam Review 350 - 360 8 MODULES The...

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Unformatted text preview: 360 8. MODULES The sequences . .1/; : : : ; .n// ..i; j / .1/; : : : ; .i; j / .n// and are identical, except that the positions of the entries i and j are reversed. Since xi D xj , the sequences .x .1/ ; : : : ; x .n/ / and .x.i;j / .1/ ; : : : ; x.i;j / .n/ / are identical. Therefore, '.x .1/ ; : : : ; x .n/ / ' .x.i;j / .1/ ; : : : ; x.i;j / .n/ / D 0: This shows that A.'/.x1 ; : : : ; xn / D 0. I For the remainder of this section, we assume R is a commutative ring with multiplicative identity. Let .a1 ; a2 ; : : : ; an / be a sequence of elements of Rn . Denote the i –th entry of aj by ai;j . In this way, the sequence .a1 ; a2 ; : : : ; an / is identified with an n–by–n matrix whose j –th column is aj . Let ' W .Rn /n ! R be the multilinear function '.a1 ; a2 ; : : : ; an / D a1;1 an;n . Define D A.'/. Thus, X . /'.a .1/ ; : : : ; a .n/ / .a1 ; : : : ; an / D 2 Sn D X (8.3.1) . /a1; an; .1/ .n/ : 2 Sn According to Lemma 8.3.5, is an alternating multilinear function. MoreO O over, satisfies .e1 ; : : : ; en / D 1. The summand belonging to in Equation 8.3.1 can be written as ./ n Y ai; .i / i D1 D. Y D./ ai;j .i;j / j D .i/ 1 / Y ai;j D . 1 / n Y a 1 .j /;j j D1 .i;j / 1 .j / iD Therefore .a1 ; : : : ; an / D X . 1 /a 1 .1/;1 a 2Sn D X 1 .n/;n : (8.3.2) . /a .1/;1 a .n/;n : 2Sn Now suppose that W .Rn /n ! N is an alternating multilinear function, where N is any R–module. Let .a1 ; a2 ; : : : ; an / be any sequence ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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